My take on it: it is a linear combination of the solution to the homogeneous equations given the boundary conditions (1) and the inhomogeneous solution (2)

(2) is simple, using the Duhamel integral or change of variables to the characteristic curves: $$ \frac{-1}{4c^2} \int _{0}^{t} \int_{x-c(t-t')}^{x+c(t-t')} f(x',t')\,dx'dt' $$

(1) is a bit confusing because of the intervals: $$ g(t) = \phi(2ct) + \psi(0), t>0 $$ $$ h(t) = \phi(0) + \psi(2ct), t<0 $$ Do we use an odd extension/ method of reflection for the functions h and g so we can solve them?